D×S×F (D×H)× H(S×F) (DH)× H(S×F) G× H(S×F) (pb) D×S (D×H)× HS m× Hid (DH)× HS G× HS \array{ D \times S \times F & \simeq & (D \times H) \times_H (S \times F) & \overset{\simeq}{\longrightarrow} & (D \cdot H) \times_H (S \times F) & \longrightarrow & G \times_H ( S \times F ) \\ \big\downarrow && && \big\downarrow & {}^{{}_{(pb)}} & \big\downarrow \\ D \times S &\simeq& (D \times H) \times_H S & \underoverset { m \times_H id } {\simeq} {\longrightarrow} & (D \cdot H) \times_H S &\hookrightarrow& G \times_H S }

Throughout, let G 1,G 2G_1, G_2 \in TopologicalGroups and consider a continuous homomorphism of topological groups

ϕ:G 1G 2. \phi \;\colon\; G_1 \longrightarrow G_2 \,.


(pullback action)

TopologicalG 1Spacesϕ *TopologicalG 2Spaces Topological G_1 Spaces \overset{ \;\;\; \phi^\ast \;\;\; }{\longleftarrow} Topological G_2 Spaces

for the functor which takes a topological G2-space (X,ρ)(X,\rho) to the same underlying topological space ρ\rho, equipped with the G 1G_1-action ρ(ϕ()) \rho(\phi(-)).


The pullback action functor (Def. ) is the left adjoint of a pair of adjoint functors

G 1SpacesMaps(G 2,) G 1ϕ *G 2Spaces G_1 Spaces \underoverset { \underset{ Maps \big( G_2, - \big)^{G_1} }{\longrightarrow} } { \overset{ \phi^\ast }{ \longleftarrow } } {\bot} G_2 Spaces

where for XTopologicalG 1SpacesX \in Topological G_1 Spaces the expression Maps(G 2,) G 1Maps(G_2,-)^{G_1} denotes the G 1G_1-fixed locus in the mapping space between topological spaces equipped with G 1G_1-actions (on G 2G_2 the ϕ\phi-induced left multiplication action) and equipped with the G 2G_2-action given by

(1)G 2×Maps(G 2,X) G 1 Maps(G 2,X) G 1 (g 2,h) h(()g 2). \array{ G_2 \times Maps(G_2,X)^{G_1} & \overset{}{\longrightarrow} & Maps(G_2,X)^{G_1} \\ (g_2, h) &\mapsto& h\big( (-) \cdot g_2 \big) \,. }


To see the defining hom-isomorphism, consider a G 1G_1-equivariant continuous function

ϕ *XfY. \phi^\ast X \overset{ \;\;\; f \;\;\; }{ \longrightarrow } Y \,.

From this we obtain the following function

X f˜ Maps(G 2,Y) G 1 x (g 2f(g 2x)), \array{ X & \overset{ \tilde f }{ \longrightarrow } & Maps \big( G_2, Y \big)^{G_1} \\ x &\mapsto& \big( g_2 \mapsto f( g_2 \cdot x ) \big) \,, }

where eG 2e \in G_2 denotes the neutral element.

This is manifestly:

  • well-defined, due to the G 1G_1-equivariance of ff;

  • continuous, being built from composition of continuous map;

  • G 2G_2-equivariant with respect to the action (1).

Conversely, given a G 2G_2-equivariant continuous function Xf˜Maps(G 2,Y) G 1X \overset{\tilde f}{\longrightarrow} Maps\big(G_2, Y\big)^{G_1}, we obtain the following function

ϕ *X Y x f˜(x)(e). \array{ \phi^\ast X &\overset{}{\longrightarrow}& Y \\ x &\mapsto& \tilde f(x)(e) \,. }

This is:

  • continuous, being the composition of continuous functions;

  • G 1G_1-equivariant due to the equivariance properties of f˜\tilde f:

    ϕ(g 1)x f˜(ϕ(g 1)x)(e) =f˜(x)(eϕ(g 1)) =f˜(x)(ϕ(g 1)e) =g 1(f˜(x)(e)) \begin{aligned} \phi(g_1) \cdot x & \mapsto \tilde f \big( \phi(g_1)\cdot x \big) (e) \\ & = \tilde f ( x ) \big( e \cdot \phi(g_1) \big) \\ & = \tilde f ( x ) \big( \phi(g_1) \cdot e \big) \\ & = g_1 \cdot \big( \tilde f ( x ) ( e ) \big) \end{aligned}

Finally, it is clear that these transformations ff˜f \leftrightarrow \tilde f are natural, hence it only remains to see that they are bijective:

Plugging in the above constructions we find indeed:

f˜˜:xf(ex)=f(x) \widetilde {\tilde f} \;\colon\; x \mapsto f(e \cdot x) = f(x)


f˜˜˜:x (g 2f˜(g 2x)(e)) =(g 2f˜(x)(eg 2)) =(g 2f˜(x)(g 2)). \begin{aligned} \widetilde {\widetilde {\tilde f}} \;\colon\; x & \mapsto \big( g_2 \mapsto \tilde f(g_2 \cdot x)(e) \big) \\ & = \big( g_2 \mapsto \tilde f(x)(e \cdot g_2) \big) \\ & = \big( g_2 \mapsto \tilde f(x)(g_2) \big) \,. \end{aligned}

G^× H^S G^× H^H^ G× HS G× HH \array{ \widehat G \times_{\widehat H} S &\longrightarrow& \widehat G \times_{\widehat H} \widehat H &{\phantom{}}& \\ \big\downarrow && \big\downarrow \\ G \times_H S &\longrightarrow& G \times_H H }
(Γ αG)× H^S (Γ αG)/H^ G× HS G/H \array{ \big( \Gamma \rtimes_\alpha G \big) \times_{\widehat H} S & \longrightarrow & \big( \Gamma \rtimes_\alpha G \big) / \widehat H \\ \big\downarrow && \big\downarrow \\ G \times_H S & \underoverset {}{}{\longrightarrow} & G/H }

(Γ αG)× H^S (Γ αG)× H^H^ G× H^S G× H^H^ \array{ \big( \Gamma \rtimes_\alpha G \big) \times_{\widehat H} S & \longrightarrow & \big( \Gamma \rtimes_\alpha G \big) \times_{\hat H} \hat H \\ \big\downarrow && \big\downarrow \\ G \times_{\hat H} S & \underoverset {}{}{\longrightarrow} & G \times_{\hat H} \widehat H }



Please do not delete the following example for the moment!

An article by vandenBergGarner2011 and another by MarraReggio2020.


  • Vincenzo Marra and Luca Reggio, A characterisation of the category of compact Hausdorff spaces, Theory Appl. Categ 35, 1871-1906 (2020) (arXiv:1808.09738)

  • Benno van den Berg and Richard Garner, Types are weak ω-groupoids, Proc. Lond. Math. Soc. (3) 102, No. 2, 370-394 (2011) (arXiv:0812.0298, doi:10.1112/plms/pdq026)

  • Jiří Adámek and Jiří Rosický, How nice are free completions of categories?, Topology Appl. 273, 24 (2020)


1348(p 214p 1 2)14(14p 1 2) \frac{13}{48} ( p_2 - \frac{1}{4} p_1^2 ) - \frac{1}{4} (\frac{1}{4} p_1^2)
1312(p 214p 1 2)(14p 1 2) \frac{13}{12} ( p_2 - \frac{1}{4} p_1^2 ) - (\frac{1}{4} p_1^2)
1312p 22512(14p 1 2) \frac{13}{12} p_2 - \frac{25}{12} (\frac{1}{4} p_1^2)
13p 25p 1 2 13 p_2 - 5 p_1^2



Q h(ϕ(h),h) H \array{ Q \\ \mathllap{ {}^{{}_{ h \mapsto (\phi(h),h) }} } \big\uparrow \\ H }
(ϕ(g 1g 2),g 1g 2)=(ϕ(g 1),g 1)(ϕ(g 2),g 2)=(ϕ(g 1)α(g 1)(ϕ(g 2)),g 1g 2) (\phi(g_1 g_2), g_1 g_2) = (\phi(g_1), g_1) \cdot (\phi(g_2), g_2) = \big( \phi(g_1) \cdot \alpha(g_1)(\phi(g_2)), g_1 \cdot g_2 \big)
(γ,u) [γ,σ(u)] Γ×U [id×σ] (Γ αG |U)/Q γϕ(σ(u)),u [γ,g] \array{ (\gamma, u) &\mapsto& [\gamma, \sigma(u)] \\ \Gamma \times U & \underoverset{}{[id \times \sigma]}{ \rightleftarrows } & \big( \Gamma \rtimes_\alpha G_{\vert U} \big)/ Q \\ \gamma \phi(\sigma(u)), u && [\gamma, g] }
[γ,g]=(γ,g)(ϕ(g 1σ(u)),g 1σ(u))=(γα(g)(ϕ(g 1σ(u)))) [\gamma,g] \;=\; (\gamma,g) \cdot ( \phi(g^{-1} \sigma(u)), g^{-1} \sigma(u) ) \;=\; ( \gamma \alpha(g)(\phi(g^{-1} \sigma(u))) )
γα(g)(ϕ(g 1σ(u)))=γα(g)(ϕ(g 1)α(g 1)(ϕ(σ(u))))=γα(g)(ϕ(g 1))ϕ(σ(u)) \gamma \alpha(g)(\phi(g^{-1} \sigma(u))) = \gamma \alpha(g)( \phi(g^{-1}) \alpha(g^{-1})(\phi(\sigma(u))) ) = \gamma \alpha(g)(\phi(g^{-1})) \phi(\sigma(u))
G× H(S×Γ) G× HΓ = (Γ×G)/(graph(HΓ)) G× HS G× HH \array{ G \times_H (S \times \Gamma) & \longrightarrow & G \times_H \Gamma & = & (\Gamma \times G) \big/ ( graph(H \to \Gamma) ) \\ \big\downarrow && \big\downarrow & \swarrow \\ G \times_H S & \underoverset {} {} {\longrightarrow} & G \times_H H }

Last revised on April 8, 2021 at 10:46:35. See the history of this page for a list of all contributions to it.